Bitcoins value does indeed affect the price of all altcoins since BTC/AltCoin trading pairs are the most commonly used and is usually the standard go to exchange pair on all cryptocurrency exchanges. So can Altcoin break all the relationship from bitcoin and run as a independent currency.
as all exchanges usually base altcoin trad price on btc, altcoins is tied on bitcoin price. however there are some fiat-pegged altcoins like usdt that have their valued tied up to fiat. also, there is this cryptocurrency called petro which has its value based on crude oil. currencies like this will not be dependent much on bitcoin's price.